Allardyce told reporters on Friday that high-flying West Ham, fifth in the table with 24 points from their opening 14 games, bid for Ivory Coast international Bony in 2013 while he was playing for Dutch team Vitesse Arnhem.
The manager choose to sign Carroll, however, in a 15 million pounds deal from Liverpool instead of the powerful Ivorian, who then completed a move to Wales.
Bony has struck six league goals this season to follow an impressive 16 goal-haul in the previous campaign while Carroll has struggled with injury since his move, making just 19 league appearances in two years and scoring twice.
The West Ham boss says his side must now stop the man he opted against signing.
"It was a case of one or the other. We couldn't do both (deals) with the budget we had," Allardyce told a news conference on Friday.
"It'll be very difficult to contain him (Bony) because he's in such good form. We'll have to limit the supply into him from the midfield and wide men. We need to get tight to him but not too tight because he can roll defenders well, he's strong and very skilful.
"To try and knock him off the ball is not an easy task. It'll be a good test for our lads to play against a player who's in such good form."
West Ham have exceeded all expectations so far in this campaign and a new brand of attacking football has appeased critics of the team's style last season.
Allardyce expects Swansea, who have also transformed themselves from last season's relegation candidates to a side challenging in the top half of the table this term, to present a tough challenge after a week in which West Ham beat Newcastle United and West Bromwich Albion.
"It's been a busy week and we've got six points so far so we're all happy with that. However I think Sunday's game is the biggest test of them all," he said.
"Three games in eight days is a big test but we're approaching it with confidence. If we win what a great week it would've been for us."
(Editing by Toby Davis)
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